Page 81 - Modern Spatiotemporal Geostatistics

62 Modern Spatiotemporal Geostatistics — Chapter 2

One may consider the metric as a transformation A = T(/II, ... ,/i n,r) of
the original coordinate system, where T has a Riemannian structure and the
forms of the coefficients gij are sought on the basis of physical and mathemat-
ical facts. Let A be expressed as

While the finite space/time distance (Eq. 2.47) has the same form as the
infinitesimal Riemannian distance (Eq. 2.27), the <jy do not necessarily coincide
with the metric coefficients of Equation 2.27. In Equation 2.47, the g^ are
assumed to denote functions of the spatial and lag distances rather than the
local coordinates, as was the case of the definition in Equation 2.27; i.e..,
9ij = 9ij(hi, hj), g 0i = g 0i(T, hi), i, j = I, . . . ,n and 500 = Soo(r). Then,
Equation 2.46 yields the expressions

The determination of the coefficient </y is not always an easy task. In most
cases it requires additional assumptions that are based on theoretical and ex-
perimental facts (e.g., the local properties of space and time, and rules imposed
by the specific natural processes). This is, clearly, an important issue that has
been discussed already in previous sections.
If the gij are space and time independent (e.g., the analysis is focused on
a local mesh), Equation 2.48 reduces to the simpler expressions below (i, j =
l,...,n)